PUH Theorem 309 (Two Timescales and a First Lower Bound) — The Encounter Rate and the Merger Duration Are Different Questions; Photon-Channel Throughput Cannot Govern Merger; and Observed Merger Times Give R_sub > 4×10²³ kg m⁻² s⁻¹

Photonic Universe Hypothesis (PUH) — Separation, Exclusion, and a Conditional Bound. THE CONFLATION. T255 states in its own status line that its merger rate λ is parametrised, not derived, and that has been read as leaving core-merger microphysics open. Two distinct quantities have been run together. (i) THE ENCOUNTER RATE: how often a small core is captured by a large one — a number density times a cross-section times a relative velocity, astrophysical rather than substrate physics. This is what T255 parametrises; its inputs (the derived core mass spectrum, T287) now exist in the archive even though the calculation has not been done. (ii) THE MERGER DURATION: how long a single merger takes once contact occurs. This is substrate physics, nothing in T255 addresses it, and it is the question worth asking of the framework. What follows concerns (ii). THEOREM 309.1 (exclusion). Suppose merger duration is throughput-limited. T213 establishes that folding and unfolding proceed at the same rate PER UNIT SURFACE AREA, so completing a merger of mass m requires processing that mass through a shell of area ~4π(GM/c²)², giving τ ~ m/(R_sub·A) = c⁴/(4πR_sub G²m). NOTE THE INVERSE MASS DEPENDENCE — larger cores merge FASTER, because shell area grows as m² while the mass grows as m. Evaluated at T294.5's upper bound on effective throughput (5.1×10¹³ kg m⁻² s⁻¹, from the persistence of Earth's core over 4.5 Gyr): a 60-solar-mass merger requires 2.4×10⁷ s ≈ nine months; a solar-mass merger 1.4×10⁹ s ≈ forty-five years; an Earth-mass core merger 5.7×10¹⁷ s — longer than the age of the universe. OBSERVED COMPACT-OBJECT MERGERS COMPLETE IN MILLISECONDS. The photon-channel throughput therefore cannot govern merger, and it fails by ten orders of magnitude rather than marginally. RESULT 309.2 (a first lower bound). The same relation inverted gives something the framework has not had. Requiring merger to complete within the observed timescale — of order ten light-crossing times of the remnant mass, three milliseconds at 60 solar masses — yields R_sub > 4.1×10²³ kg m⁻² s⁻¹. EVERY PREVIOUS CONSTRAINT ON THIS QUANTITY HAS BEEN AN UPPER BOUND: T294.5's Earth-core ceiling, and T210's Planck rate as universal ultraviolet ceiling (≈1.6×10¹⁰⁵ in these units). This is the FIRST CONSTRAINT FROM BELOW, and it arrives from gravitational-wave timing rather than planetary interiors — an independent channel. Relative to the Planck ceiling it is weak, some eighty-one orders below, but a two-sided constraint is a different kind of object from a one-sided one. THE BOUND IS CONDITIONAL on the premise of Theorem 309.1 — that merger duration is throughput-limited at all — and Result 309.3 supplies a test. THE TWO BOUNDS FORCE A SEPARATION THE ARCHIVE DECLINED TO MAKE. Result 309.2 and T294.5's ceiling differ by a factor ≈8×10⁹. Stated flatly they are inconsistent. They are reconciled by a distinction T294.5 explicitly declined to draw: it bounded an EFFECTIVE throughput with the geometric efficiency of photon delivery to a planet-embedded core folded in rather than separated out — it said so. The two coexist provided that efficiency is below ≈1.2×10⁻¹⁰. Whether an escape-cone factor for photons reaching a micron-scale core under six thousand kilometres of rock is that small is not obvious, and is not computed here. WHAT THE NUMBERS ESTABLISH IS THAT THE EFFECTIVE AND BARE THROUGHPUTS CAN NO LONGER BE TREATED AS INTERCHANGEABLE: one is bounded above at 5.1×10¹³ and the other, if merger is throughput-limited, below at 4.1×10²³. They cannot be the same number. This is a constraint on the framework rather than a contradiction within it, and it makes the photon-delivery efficiency for embedded cores newly load-bearing. RESULT 309.3 (the mass scaling is the discriminant). Two possibilities remain, distinguished by mass dependence rather than magnitude — which makes the test independent of the throughput's unknown value. THROUGHPUT-LIMITED: duration scales inversely with mass, so SMALL CORES MERGE SLOWEST; at the throughput Result 309.2 demands, an Earth-mass core merger takes ≈7×10⁷ s (about two years) while a 60-solar-mass merger completes in milliseconds — an enormous spread across the astrophysical range. TOPOLOGICAL: T172 establishes that two Casimir leaves merging into one is topologically allowed while splitting is forbidden, and A TOPOLOGICAL OPERATION CARRIES NO RATE; merger would be instantaneous at every mass with no mass dependence whatsoever. THE ARCHIVE'S OWN THEOREM LEANS TOWARD THE SECOND: T172 states leaf-merging as a fact about the foliation, not as a transport process, and reading it as rate-limited flow imports a dynamical character the theorem does not assert. If that reading is correct, Theorem 309.1's exclusion stands but Result 309.2's bound does not follow, because merger duration would carry no information about throughput at all. A REMARK ON THE ARENA: T156's Casimir phase structure places an entire phase between the critical value c* defining the Shell and the maximum attained at the core; a merging configuration must traverse it, since two shells at c* cannot become one shell at c* without passing through intermediate values. That is where any dynamical account of merger would have to live. KILL-CONDITIONS: (i) if the shell area relevant to throughput is not ~4π(GM/c²)² — for instance if the Shell radius carries the dimensionless threshold ratio as a substantial factor — the coefficients shift, though a ten-order discrepancy cannot be removed by an order-unity factor; (ii) if the photon-delivery efficiency for embedded cores exceeds 1.2×10⁻¹⁰, the two bounds are genuinely inconsistent and one must fail; (iii) if merger duration is topological, Result 309.2's bound is void; (iv) if merging cores are ever resolved across a wide mass range with no mass dependence in duration, the throughput reading is excluded and the topological one confirmed. NOT CLAIMED: that the encounter rate has been derived; that merger is throughput-limited (309.1 excludes one value, not the mechanism); that the lower bound is unconditional; that the photon-delivery efficiency has been computed; that the transient re-equilibration accompanying merger has been described, only that the phase structure supplies its arena; or that the substrate throughput has been derived from anything, which remains the standing open problem.